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Compressive Sensing Universidade Federal do Rio de Janeiro Compressive Sensing Claudio Mayrink Verdun Rio de Janeiro Dezembro de 2016 i Universidade Federal do Rio de Janeiro Compressive Sensing Claudio Mayrink Verdun Dissertação de Mestrado apresentada ao Programa de Pós-graduação em Matemática Aplicada, In- stituto de Matemática da Universidade Federal do Rio de Janeiro (UFRJ), como parte dos requisitos necessários à obtenção do título de Mestre em Matemática Aplicada. Orientador: Prof. Cesar Javier Niche Mazzeo. Co-orientador: Prof. Bernardo Freitas Paulo da Costa. Co-orientador: Prof. Eduardo Antonio Barros da Silva. Rio de Janeiro Dezembro de 2016 iii Verdun, Claudio Mayrink V487c Compressive Sensing / Claudio Mayrink Verdun. – Rio de Janeiro: UFRJ/ IM, 2016. 202 f. 29,7 cm. Orientador: César Javier Niche Mazzeo. Coorientadores: Bernardo Freitas Paulo da Costa, Eduardo Ântonio Barros da Silva. Dissertação (Mestrado) — Universidade Federal do Rio de Janeiro, Instituto de Matemática, Programa de Pós-Graduação em Matemática, 2016. 1. Compressive Sensing. 2. Sparse Modelling. 3. Signal Processing. 4. Applied Harmonic Analysis. 5. Inverse Problems I. Mazzeo, César Javier Niche, orient. II. Costa, Bernardo Freitas Paulo da, coorient. Silva, Eduardo Ântonio Barros da, coorient. III. Título iv v Del rigor en la ciencia En aquel Imperio, el Arte de la Cartografía logró tal Perfección que el Mapa de una sola Provincia ocupaba toda una Ciudad, y el Mapa del Imperio, toda una Provincia. Con el tiempo, estos Mapas Desmesurados no satisficieron y los Colegios de Cartógrafos levantaron un Mapa del Imperio, que tenía el Tamaño del Imperio y coincidía puntualmente con él. Menos Adictas al Estudio de la Cartografía, las Generaciones Siguientes entendieron que ese dilatado Mapa era Inútil y no sin Impiedad lo entregaron a las Inclemencias del Sol y los Inviernos. En los Desiertos del Oeste perduran despedazadas Ruinas del Mapa, habitadas por Animales y por Mendigos; en todo el País no hay otra reliquia de las Disciplinas Geográficas. Suárez Miranda: Viajes de varones prudentes Libro Cuarto, cap. XLV, Lérida, 1658. In [Borges ’60]. vi Acknowledgments This is the only part of my dissertation written in my native language. I hope that all who are cited here may read clearly what follows. Se você é daquelas pessoas que só liga para a matemática e acha que ela não é feita por seres humanos que sorriem, choram, pensam coisas profundas mas também fazem coisas triviais, pule esta parte e vá diretamente para o Capítulo 1. Ali, vetores esparsos e medições compressivas te esperam. Eu tentei ser preciso e não esquecer ninguém. Se eu esqueci, possivelmente foi de propósito, se não foi, peço desculpas, embora nunca irá transparecer qual foi o motivo do esquecimento. Além disso, veremos ao longo dessa dissertação que redundância é algo fundamental, então começaremos por aqui, citando algumas pessoas em mais de um lugar e por motivos diferentes. Esta dissertação é dedicada a duas santíssimas trindades: A primeira delas, Felipe Acker, o pai, Fábio Ramos, o filho, e Luiz Wagner Biscainho, o espírito santo. A segunda, Roberto Velho, o pai, Filipe Goulart, o filho, e Hugo Carvalho, o espírito santo. Essas seis pessoas contribuíram de maneira não trivial para minha formação e introduziram muitas não-linearidades na minha vida ao longo dos últimos anos. A quantidade de coisas que elas me ensinaram e todas as vezes que fui ajudado por elas, mesmo que as três horas da manhã, as vezes desesperado, e com pedidos nada convencionais não pode ser descrito aqui em poucas palavras. Parte substancial da minha formação humana e científica deve-se aos laços estreitos e inúmeras conversas que mantenho com esses seis indi- víduos bastante singulares. Começo agradecendo três professores que tive na escola e que foram fonte de inspiração para a escolha da carreira científica: Andre Fontenele, José Carlos de Medeiros e José Alexandre Tostes Lopes (in memoriam). Aproveitei muitos os três anos de contato com eles, as conversas que iam de geometria euclidiana à suma teológica de São Tomás de Aquino e todos os ensinamentos sobre porque o mundo era muito maior que o IME/ITA e porque haviam opções melhores sobre o que fazer da vida. Agradeço ao Vinícius Gripp por ter me mostrado a sociedade secreta que é a matemática aplicada da UFRJ. Se eu não tivesse passado mal um certo dia na escola e não tivesse precisado ir embora, não teria conhecido o Vinícius, todo o seu entusiasmo e tudo o que se seguiu a partir daí. Além disso, agradeço aos seis bandeirantes da matemática aplicada: Enio Hayashi, Pedro Maia, Rafael March, Roberto Velho, Rodrigo Targino e Yuri Saporito. Esses veteranos, que acabaram virando amigos, continuam por aí desbravando a mata fechada e eu sou muito feliz em aproveitar muitos dos conselhos por eles oferecidos. Aos vários mestres que tive nos últimos sete anos. Cada lição, cada raciocínio, cada forma de fazer as coisas dentro e fora da sala de aula me mostrou um mundo de ideias interessantes que eu ainda vou utilizar muito nas minhas pesquisas e atividades didáticas. São eles Adan Corcho, Alexander Arbieto, Alexandra Schmidt, Amit Bhaya, Átila Pantaleão, Bernardo da Costa, Bruno Scárdua, César Niche, Didier Pilod, Eduardo Barros Silva, Fábio Ramos, Felipe Acker, Flávio Dickstein, Heudson Mirandola, Luca Moriconi, Marco Aurélio Cabral, Mar- cus Venícius Pinto, Luiz Wagner Biscainho, Nilson Bernardes, Paulo Diniz, Sergio Lima Netto, Severino Collier, Umberto Hryniewicz, Wagner Coelho, Wallace Martins e Wladimir Neves. Agradeço também à vários outros professores que conheci na universidade e que me ensinaram como não fazer as coisas, como não trabalhar, como não fazer pesquisa e dar aula, como não tratar os alunos, etc. Aos muitos amigos e colegas que fiz aplicada e na UFRJ. Alguns se tornaram muito próximos, outros nem tanto, mas todos eles compartilharam momentos felizes e ajudaram a superar momentos tristes na universidade. Também me proporcionaram conversas fantásticas e muitas vezes me mostraram que eu estava errado, por mais que que fosse difícil dar o braço a torcer. São eles: Aloizio Macedo, Arthur Mitrano, Bruno Almeida, Bruno Oliveira, Bruno Santiago, Camilla Codeço, Carlos Lechner, Carlos Zanini, Cecília Mondaini, Claudio Heine, Daniel Soares, Danilo Barros, Davi Obata, Felipe Pagginelli, Felipe Senra, Flávio Ávila, Franklin Maia, Gabriel Castor, Gabriel Sanfins, Gabriel Vidal, Gabriela Lewenfus, Gabriella Radke, Ian Martins, Jhonatas Afradique, Leandro Machado, Leonardo Assumpção, Lucas Barata, Lucas Manoel, Lucas Stolerman, Marcelo Ribeiro, Marco Aurélio Soares, Milton Nogueira, Nicolau Sarquis, Paulo Cardozo, Pedro Fonini, Pedro Schmidt, Philipe de Fab- ritiis, Rafael Teixeira, Raphael Lourenço, Reinaldo de Melo e Souza, Rogério Lourenço, Thiago Degenring, Tiago Domingues, Tiago Vital, Washington Reis, Victor Coll e Victor Corcino. vii Aos top’s, a galera realmente truuu, que se tornaram super amigos e que sei poder contar para qualquer coisa em qualquer hora e lugar. Alguns deles estão longe mas sei que caso eu precise, eles não medirão esforços para estar perto. Obrigado Adriano Côrtes, Bruno Braga, Bruno Santiago, Cláudio Gustavo Lima, Cristiane Krug, Douglas Picciani, Guilherme Sales, Luís Felipe Velloso e Rafael Ribeiro por serem quem vocês são e pela amizade tão forte. Aos meus amigos de infância Hugo Portuita, Pedro Goñi e João Paulo Siqueira que sempre me procuram, insistem em me trazer para um mundo distante do acadêmico, e me lembram que o samba não pode morrer. Obrigado pelo carinho e amizade ao longo deste muitos anos. E desculpem pelo sumiço. Minhas ausências estão parcialmente justificada nas páginas que seguem. Aos meus amigos de conversas estranhas, em qualquer horário e sobre qualquer assunto, sempre regadas a muitas risadas, gritos e discussões intensas. Algumas delas, no batista às cinco da tarde, outras no 485 lotado às dez da noite que faziam as pessoas olharem com vergonha alheia para nós, seja falando de política, religião, mal dos outros ou qualquer outra coisa. Ou mesmo nos infinitos áudios do Whatsapp. Espero ainda continuar brig- ando e conversando com Adriano Côrtes, André Saraiva, Bruno Santiago, Bruno Braga, Camila Codeço, Carlos Lechner, Cristiane Krug, Daniel Soares, Douglas Picciani, Filipe Goulart, Flávio Ávila, Gabriel Vidal, Guilherme Sales, Hugo Carvalho, Luís Felipe Velloso, Nicolau Sarquis, Reinaldo de Melo e Souza, Tiago Domingues e Tiago Vital. Aos meus amigos do SMT, do lado aplicado da força. Isabela Apolinário e Renam Castro me ajudaram nas matérias com questões de computação e a entender a linguagem falada pelos engenheiros que soava tão estranha para mim. Ainda me assusto um pouco com um banco de filtros mas isso vai passar... Nossos trabalhos em equipe e seminários foram divertidos, por mais que muitas vezes eu não soubesse direito o que eu estava fazendo. Ao Flávio Ávila, grande amigo de seminários e de conversas político-ecônomicas, pela amizade aplicada. Espero finalmente ter tempo para nosso projetos começarem a andar. Meus colegas Hamed Yazdanpanah, Pedro Fonini e Rafael Zambrano, que trabalharam comigo no projeto do CENPES, me ajudaram a desbravar um mundo de engenharia de reservatórios completamente desconhecido para todos nós. Juro que tentarei não perturbar mais vocês as quatro da manhã para terminarmos um monte de relatórios e simulações. Aos professores Eduardo Barros Silva, Paulo Diniz e Sergio Lima Netto que compõem esta equipe e me ofereceram a oportunidade de trabalhar neste projeto, ajudando-me financeira e intelectualmente. A maneira como vocês encaram as situações de forma descontraída mas ao mesmo tempo profissional me encanta e fez com que todas as tarde de quarta fossem sempre engraçadas e leves mas ao mesmo tempo profundas. Aos professores Flávio Dickstein e Paulo Goldfeld por gen- tilmente cederem o espaço que utilizei para a escrita desta dissertação e para o trabalho no projeto nos últimos tempos. Foi muito interessante interagir com eles, mesmo que brevemente, e vê-los trabalhando com entusiasmo todos os dias. E ao Renan Vicente, pelas risadas, inúmeros favores com rodar simulações no IMEX para mim e pela companhia diária no LABMA.
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